Introduction

Relevance of the Theme

In mathematics, divisibility is a fundamental concept. The study of divisibility criteria is a cornerstone for the understanding of many other topics in the curriculum, such as factorization, prime numbers, and simplification of fractions. Moreover, it is one of the first explorations within the broader topic of Arithmetic, which is the basis for many other studies in mathematics.

Contextualization

The theme of Divisibility Criteria is at the heart of the Mathematics curriculum for the 1st year of High School. It is a natural extension of the topics of division and multiples, which are addressed in the 9th year of Elementary School, and serves as a preparation for more advanced mathematical concepts that will be introduced later, such as Number Theory and Algebra. A solid understanding of these criteria is essential for success in future topics, which is why this theme is so vital.

Theoretical Development

Components

• Divisibility Criteria for 2 and 5: These criteria are fundamental and are based on the properties of the final digits. If a number ends in 0, 2, 4, 6, or 8, it is divisible by 2. If it ends in 0 or 5, it is divisible by 5. These criteria are the basis for divisibility by 10 (since 10 is the product of 2 and 5).

• Divisibility Criteria for 3: The sum of the digits of a number indicates whether it is divisible by 3 or not. If the sum is divisible by 3, then the number is also divisible by 3.

• Divisibility Criteria for 4: This criterion is based on the last two digits of a number. If this pair of digits forms a number divisible by 4, then the original number is also divisible by 4.

• Divisibility Criteria for 6: For a number to be divisible by 6, it needs to be divisible by 2 and 3 at the same time. This means it ends in an even digit and the sum of its digits is divisible by 3.

• Divisibility Criteria for 9: Similar to the divisibility criterion for 3, this criterion is also determined by the sum of its digits. If the sum is divisible by 9, then the number is also divisible by 9. This criterion is derived from the principle of additivity.

• Divisibility Criteria for 10: If a number ends in 0, then it is divisible by 10. This criterion is a direct extension of the divisibility criteria for 2 and 5.

Key Terms

• Divisibility: The ability of a number to be divided by another resulting in an exact division, without remainder.
• Divisibility Criteria: Set of rules that help determine if a number is divisible by another without performing the division itself.

Examples and Cases

1. Divisibility by 2: The number 546 is divisible by 2, as it ends in an even digit. However, the number 573 is not divisible by 2, as it ends in an odd digit.

2. Divisibility by 3: The number 327 is divisible by 3, as the sum of its digits (3 + 2 + 7) is 12, which is divisible by 3. The number 956 is not divisible by 3, as the sum of its digits (9 + 5 + 6) is 20, which is not divisible by 3.

3. Divisibility by 4: The number 148 is divisible by 4, as the last two digits form the number 48, which is divisible by 4. The number 327 is not divisible by 4, as the last two digits form the number 27, which is not divisible by 4.

4. Divisibility by 6: The number 732 is divisible by 6, as it ends in an even digit and the sum of its digits (7 + 3 + 2) is 12, which is divisible by 3. The number 956 is not divisible by 6, as it does not end in an even digit.

5. Divisibility by 9: The number 4,968 is divisible by 9, as the sum of its digits (4 + 9 + 6 + 8) is 27, which is divisible by 9. The number 854 is not divisible by 9, as the sum of its digits (8 + 5 + 4) is 17, which is not divisible by 9.

6. Divisibility by 10: The number 2,540 is divisible by 10, as it ends in 0. The number 321 is not divisible by 10, as it does not end in 0.

Detailed Summary

Key Points

1. Divisibility Criteria: There are 6 criteria: divisibility by 2, 3, 4, 5, 6, and 9. They are based on properties of the final digits and the sum of the digits.

2. Divisibility by 2: A number is divisible by 2 if its final digit is 0, 2, 4, 6, or 8. This is because all multiples of 2 are even.

3. Divisibility by 5: A number is divisible by 5 if it ends in 0 or 5. This is because all multiples of 5 end in one of these two digits.

4. Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3. This is because 10-1=9, so if the sum of the digits is a multiple of 3, then the number is also.

5. Divisibility by 4: A number is divisible by 4 if the number formed by its last two digits is divisible by 4. This is because 100-4=96, so if a number where the last two digits form a multiple of 4, then the original number is also.

6. Divisibility by 6: A number is divisible by 6 if it is even and the sum of its digits is divisible by 3. This is because if it is even, it is already divisible by 2 and if the sum of the digits is a multiple of 3, then it is also divisible by 3.

7. Divisibility by 9: A number is divisible by 9 if the sum of its digits is divisible by 9. This is because 10-1=9, therefore, if the sum of the digits is a number divisible by 9, then the original number is also.

8. Divisibility by 10: A number is divisible by 10 if its final digit is 0. This is because 10 is the product of 2 and 5, and any number divisible by 2 and 5 will also be divisible by 10.

Conclusions

1. Understanding the divisibility criteria is essential for simplifying division operations and for number analysis. Knowing if a number is divisible by another without the need to perform the division itself saves time and effort.

2. The divisibility criteria are a direct consequence of the addition and multiplication properties of numbers. Therefore, they are logically consistent and apply to any number, regardless of how large or small it is.

3. The divisibility criteria provide an efficient way to verify if a number has been correctly divided. By checking each criterion, one can confirm if the division was done correctly, without the need to redo the division.

Exercises

1. Check if the numbers 235, 480, 29,625, and 9,312 are divisible by 2, 5, 3, 4, 6, 9, and 10.

2. Using the divisibility criteria, determine if the following numbers are divisible by 100: 348,000; 1,892; 5,500. What are the multiples of 100 that are less than these numbers?

3. Which numbers, less than 1000, are divisible by 2, 3, 4, 5, 6, 9, and 10? How many are there in total?